Difference between revisions of "2004 AMC 10A Problems/Problem 19"

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A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
 
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
  
[[Image:2004 AMC 10A problem 19.png]]
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<asy>
 
+
size(250);defaultpen(linewidth(0.8));
 +
draw(ellipse(origin, 3, 1));
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fill((3,0)--(3,2)--(-3,2)--(-3,0)--cycle, white);
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draw((3,0)--(3,16)^^(-3,0)--(-3,16));
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draw((0, 15)--(3, 12)^^(0, 16)--(3, 13));
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filldraw(ellipse((0, 16), 3, 1), white, black);
 +
draw((-3,11)--(3, 5)^^(-3,10)--(3, 4));
 +
draw((-3,2)--(0,-1)^^(-3,1)--(-1,-0.89));
 +
draw((0,-1)--(0,15), dashed);
 +
draw((3,-2)--(3,-4)^^(-3,-2)--(-3,-4));
 +
draw((-7,0)--(-5,0)^^(-7,16)--(-5,16));
 +
draw((3,-3)--(-3,-3), Arrows(6));
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draw((-6,0)--(-6,16), Arrows(6));
 +
draw((-2,9)--(-1,9), Arrows(3));
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label("$3$", (-1.375,9.05), dir(260), fontsize(7));
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label("$A$", (0,15), N);
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label("$B$", (0,-1), NE);
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label("$30$", (0, -3), S);
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label("$80$", (-6, 8), W);</asy>
 
<math> \mathrm{(A) \ } 120 \qquad \mathrm{(B) \ } 180 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 360 \qquad \mathrm{(E) \ } 480  </math>
 
<math> \mathrm{(A) \ } 120 \qquad \mathrm{(B) \ } 180 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 360 \qquad \mathrm{(E) \ } 480  </math>
  

Revision as of 14:46, 31 December 2011

Problem

A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

[asy]  size(250);defaultpen(linewidth(0.8)); draw(ellipse(origin, 3, 1)); fill((3,0)--(3,2)--(-3,2)--(-3,0)--cycle, white); draw((3,0)--(3,16)^^(-3,0)--(-3,16)); draw((0, 15)--(3, 12)^^(0, 16)--(3, 13)); filldraw(ellipse((0, 16), 3, 1), white, black); draw((-3,11)--(3, 5)^^(-3,10)--(3, 4)); draw((-3,2)--(0,-1)^^(-3,1)--(-1,-0.89)); draw((0,-1)--(0,15), dashed); draw((3,-2)--(3,-4)^^(-3,-2)--(-3,-4)); draw((-7,0)--(-5,0)^^(-7,16)--(-5,16)); draw((3,-3)--(-3,-3), Arrows(6)); draw((-6,0)--(-6,16), Arrows(6)); draw((-2,9)--(-1,9), Arrows(3)); label("$3$", (-1.375,9.05), dir(260), fontsize(7)); label("$A$", (0,15), N); label("$B$", (0,-1), NE); label("$30$", (0, -3), S); label("$80$", (-6, 8), W);[/asy] $\mathrm{(A) \ } 120 \qquad \mathrm{(B) \ } 180 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 360 \qquad \mathrm{(E) \ } 480$

Solution

The cylinder can be "unwrapped" into a rectangle, and we see that the stripe is a parallelogram with base 3 and height 80. $3\cdot 80=240\Rightarrow \boxed{\mathrm{(C) \ }}$

See also

2004 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 18
Followed by
Problem 20
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All AMC 10 Problems and Solutions